Problem: The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\ldots,r_{2007}$.  A polynomial $P$ of degree $2007$ has the property that
\[P\left(r_j+\dfrac{1}{r_j}\right)=0\]for $j=1,\ldots,2007$.  Determine the value of $\frac{P(1)}{P(-1)}$.
Explanation: We can write
\[f(x) = (x - r_1)(x - r_2) \dotsm (x - r_{2017})\]and
\[P(z) = k \prod_{j = 1}^{2007} \left( z - \left( r_j + \frac{1}{r_j} \right) \right)\]for some nonzero constant $k.$

We want to compute
\[\frac{P(1)}{P(-1)} = \frac{\prod_{j = 1}^{2007} \left( 1 - \left( r_j + \frac{1}{r_j} \right) \right)}{\prod_{j = 1}^{2007} \left( -1 - \left( r_j + \frac{1}{r_j} \right) \right)} = \frac{\prod_{j = 1}^{2007} (r_j^2 - r_j + 1)}{\prod_{j = 1}^{2007} (r_j^2 + r_j + 1)}.\]Let $\alpha$ and $\beta$ be the roots of $x^2 + x + 1 = 0,$ so
\[x^2 + x + 1 = (x - \alpha)(x - \beta).\]Then
\[x^2 - x + 1 = (x + \alpha)(x + \beta).\]Also, $(\alpha - 1)(\alpha^2 + \alpha + 1) = \alpha^3 - 1 = 0,$ so $\alpha^3 = 1.$  Similarly, $\beta^3 = 1.$  Thus,
\begin{align*}
\prod_{j = 1}^{2007} (r_j^2 - r_j + 1) &= \prod_{j = 1}^{2007} (r_j + \alpha)(r_j + \beta) \\
&= \prod_{j = 1}^{2007} (-\alpha - r_j)(-\beta - r_j) \\
&= f(-\alpha) f(-\beta) \\
&= (-\alpha^{2007} + 17 \alpha^{2006} + 1)(-\beta^{2007} + 17 \beta^{2006} + 1) \\
&= (17 \alpha^2)(17 \beta^2) \\
&= 289.
\end{align*}Similarly,
\begin{align*}
\prod_{j = 1}^{2007} (r_j^2 + r_j + 1) &= \prod_{j = 1}^{2007} (r_j - \alpha)(r_j - \beta) \\
&= \prod_{j = 1}^{2007} (\alpha - r_j)(\beta - r_j) \\
&= f(\alpha) f(\beta) \\
&= (\alpha^{2007} + 17 \alpha^{2006} + 1)(\beta^{2007} + 17 \beta^{2006} + 1) \\
&= (17 \alpha^2 + 2)(17 \beta^2 + 2) \\
&= 289 \alpha^2 \beta^2 + 34 \alpha^2 + 34 \beta^2 + 4 \\
&= 259.
\end{align*}Therefore,
\[\frac{P(1)}{P(-1)} = \boxed{\frac{289}{259}}.\]